Fractional parts of Dedekind sums in function fields
The paper under review considers Dedekind sums in function fields, and shows some of the properties of the classical Dedekind sums also hold. Let \({\mathbb F}_q\) be the finite field with \(q\) elements, \(A={\mathbb F}_q[T]\) with \(T\) an indeterminate and \(K={\mathbb F}_q(T)\). Also let \(K_{\infty}\) be the completion of \(K\) at \(\infty=(1/T)\) and \(C_{\infty}\) be the completion of an algebraic closure of \(K_{\infty}\). For a rank \(r\) \(A\)-lattice \(\Lambda\), i.e., a finitely generated \(A\)-submodule \(\Lambda\) of rank \(r\) in \(C_{\infty}\) that is discrete in the topology of \(C_{\infty}\), there exist a Drinfeld \(A\)-module \(\phi^{\Lambda}\) of rank \(r\) over \(C_{\infty}\) and a surjective analytic function \(e_{\Lambda}\) \(:C_{\infty}\rightarrow C_{\infty}\) such that \[ e_{\Lambda}(az)=\phi_a^{\Lambda}(e_{\Lambda}(z))\;\;\text{for all}\;\; a\in A\; \] with some other properties. The Dedekind sum associated with \(\Lambda\) is defined for \(a,\,c\in A\setminus \{0\}\) with \((a,c)=1\) by \[ s_{\Lambda}(a,c)=\frac{1}{c}\sum_{0\neq \lambda\in \Lambda/c\Lambda}e_{\Lambda} \left(\frac{\lambda}{c}\right)^{-1}e_{\Lambda}\left(\frac{a\lambda}{c}\right)^{-1}\;. \] This is an analogue of the classical Dedekind sum, and the reciprocity law has been obtained for more general sums in [\textit{A. Bayad} and the author, Acta Arith. 152, No. 1, 71--80 (2012; Zbl 1301.11047)]. For a rank 1 \(A\)-lattice \(L\) which corresponds to the Carlitz module and for \(q=3\) or 2, the normalized Dedekind sum is defined as \[ {\mathcal S}(a,c)=\begin{cases}(T^3-T)s_{L}(a,c)\qquad &\text{if}\;\; q=3\,,\\ (T^4+T^2)s_{L}(a,c)\quad &\text{if}\;\;q=2\;. \end{cases} \] In this paper, the sums \({\mathcal S}(a,c)\) are investigated, and there are three main results: {\parindent=0.7cm \begin{itemize} \item[(i)] Fractional parts of the \({\mathcal S}(a,c)\) are determined~(Theorem 3.1). Also it is shown that any element of \(K\) can be a fractional part of \({\mathcal S}(a,c)\) for some \(a,\,c\in A\setminus\{0\}\)~(Theorem 3.2); \item [(ii)] The Rademacher function \(\Phi\,:\mathrm{SL}_2(A)\rightarrow A\) is defined by using (i), which is an analogue of the Rademacher function coming from the Dedekind \(\eta\)-function, and Theorem 4.2 states that \(\Phi\) is a group homomorphism; \item [(iii)] The three term relation for \({\mathcal S}(a,c)\) is obtained~(Theorem 4.7), which is proved by using \(\Phi\). \end{itemize}}
scientific article; zbMATH DE number 6608237
scientific article